On An Unstable Thin-film Equation With Convective Term

We studied the initial boundary value problem of the unstable thin-film equation (?) where Ω(?)RN is an arbitrary bounded domain(N ≥ 1)with boundary(?)Ω of the class C1.1,QT =(0,T)×Ω,n>0,m∈R,a0>0,and a1 ∈ R.Because of the degeneracy,we first considered the regularized problem ht + div(fγ(h))(a0▽△h +a1Dε"(h)▽h)=(?)·▽B(h),在 QT,h = △h = 0,在(?)Ω(0,T),h(x,0)= h0,δε(x),where (?)Using the Faedo-Galerkin method,we proved the existence of solutions for the regularized parabolic problem.By means of energy and entropy estimates,using the Poincare inequality and the Young inequality,we proved the existence of non-negative weak solutions and strong solutions for the initial boundary value problem.